Abstract

A single qubit may be represented on the Bloch sphere or similarly on the
$3$-sphere $S^3$. Our goal is to dress this correspondence by converting the
language of universal quantum computing (uqc) to that of $3$-manifolds. A magic
state and the Pauli group acting on it define a model of uqc as a POVM that one
recognizes to be a $3$-manifold $M^3$. E. g., the $d$-dimensional POVMs defined
from subgroups of finite index of the modular group $PSL(2,\mathbb{Z})$ in
\cite{PlanatModular} correspond to $d$-fold $M^3$- coverings of the trefoil
knot. In this paper, one also investigates quantum information on a few \lq
universal' knots and links such as the figure-of-eight knot, the Whitehead link
and Borromean rings \cite{Hilden1987}, making use of the catalog of platonic
manifolds available on SnapPy \cite{Fominikh2015}. Further connections between
POVMs based uqc and $M^3$'s obtained from Dehn fillings are explored.

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